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Let $R_1,R_2$ be two (left) artinian rings (not necessarily commutative), is $R_1\times R_2$ necessarily artinian ?

I also have another related question that came to my head while thinking about the first one. If $R_1,R_2$ both have a finite number of left ideals, must $R_1\times R_2$ have a finite number of left ideals too ?

Thank you

moora
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1 Answers1

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The answer to both questions is yes because the left ideals of $R_1\times R_2$ have the form $I_1\times I_2$ where $I_i$ is a left ideal of $R_i$.

This is not difficult to prove: for $I\lhd R$, just consider the left ideals $R(1,0)I$ and $R(0,1)I$.

rschwieb
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